[58f41fe] | 1 | #barbell model |
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| 2 | # cylinder model |
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| 3 | # Note: model title and parameter table are inserted automatically |
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| 4 | r""" |
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| 5 | |
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| 6 | Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of |
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| 7 | the cylinder, *L*, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than |
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| 8 | that of the cylinder and the center of the end cap radius lies outside of the cylinder. All dimensions of the BarBell |
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| 9 | are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. |
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| 10 | |
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| 11 | Definition |
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| 12 | ---------- |
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| 13 | |
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| 14 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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| 15 | |
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| 16 | The barbell geometry is defined as |
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| 17 | |
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| 18 | .. image:: img/image105.jpg |
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| 19 | |
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| 20 | where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. |
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| 21 | |
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| 22 | Since the end cap radius |
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| 23 | *R* >= *r* and by definition for this geometry *h* < 0, *h* is then defined by *r* and *R* as |
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| 24 | |
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| 25 | *h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`) |
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| 26 | |
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| 27 | The scattered intensity *I(q)* is calculated as |
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| 28 | |
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| 29 | .. math:: |
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| 30 | |
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| 31 | I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right> |
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| 32 | |
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| 33 | where the amplitude *A(q)* is given as |
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| 34 | |
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| 35 | .. math:: |
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| 36 | |
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| 37 | A(Q) =&\ \pi r^2L |
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| 38 | {\sin\left(\tfrac12 QL\cos\theta\right) |
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| 39 | \over \tfrac12 QL\cos\theta} |
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| 40 | {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\ |
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| 41 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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| 42 | \cos\left[ Q\cos\theta |
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| 43 | \left(Rt + h + {\tfrac12} L\right)\right] |
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| 44 | \times (1-t^2) |
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| 45 | {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right] |
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| 46 | \over QR\sin\theta \left(1-t^2\right)^{1/2}} |
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| 47 | |
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| 48 | The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form |
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| 49 | factor, *P(q)*. The scale factor is equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast is |
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| 50 | the difference of scattering length densities of the cylinder and the surrounding solvent. |
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| 51 | |
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| 52 | The volume of the barbell is |
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| 53 | |
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| 54 | .. math:: |
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| 55 | |
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| 56 | V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right) |
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| 57 | |
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| 58 | |
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| 59 | and its radius-of-gyration is |
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| 60 | |
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| 61 | .. math:: |
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| 62 | |
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| 63 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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| 64 | + R^4\left(6h+\tfrac32 L\right) |
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| 65 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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| 66 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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| 67 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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| 68 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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| 69 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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| 70 | |
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| 71 | **The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. |
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| 72 | |
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| 73 | This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, *qmin* = 0.001 |Ang^-1|, |
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| 74 | *qmax* = 0.7 |Ang^-1| and the following default values |
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| 75 | |
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| 76 | ============== ======== ============= |
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| 77 | Parameter name Units Default value |
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| 78 | ============== ======== ============= |
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| 79 | scale None 1.0 |
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| 80 | len_bar |Ang| 400.0 |
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| 81 | rad_bar |Ang| 20.0 |
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| 82 | rad_bell |Ang| 40.0 |
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| 83 | sld_barbell |Ang^-2| 1.0e-006 |
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| 84 | sld_solv |Ang^-2| 6.3e-006 |
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| 85 | background |cm^-1| 0 |
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| 86 | ============== ======== ============= |
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| 87 | |
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| 88 | .. image:: img/image110.jpg |
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| 89 | |
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| 90 | *Figure. 1D plot using the default values (w/256 data point).* |
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| 91 | |
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| 92 | For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for |
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| 93 | |theta| = 45 deg and |phi| = 0 deg with default values for other parameters |
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| 94 | |
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| 95 | .. image:: img/image111.jpg |
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| 96 | |
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| 97 | *Figure. 2D plot (w/(256X265) data points).* |
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| 98 | |
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[3dc41df] | 99 | .. image:: img/orientation.jpg |
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[58f41fe] | 100 | |
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[3dc41df] | 101 | Figure. Definition of the angles for oriented 2D barbells. |
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[58f41fe] | 102 | |
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[3dc41df] | 103 | .. image:: img/orientation2.jpg |
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[58f41fe] | 104 | |
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[3dc41df] | 105 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
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[58f41fe] | 106 | |
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| 107 | REFERENCE |
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| 108 | |
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| 109 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
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| 110 | |
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| 111 | H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
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| 112 | |
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| 113 | """ |
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| 114 | from numpy import pi, inf |
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| 115 | |
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| 116 | name = "barbell" |
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| 117 | title = "Cylinder with spherical end caps" |
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| 118 | description = """ |
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| 119 | Calculates the scattering from a barbell-shaped cylinder. That is a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap |
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| 120 | radius lies outside of the cylinder. |
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| 121 | Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when rad_bar = rad_bell, it is a spherocylinder. |
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| 122 | It must be that rad_bar <(=) rad_bell. |
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| 123 | """ |
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| 124 | |
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| 125 | parameters = [ |
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| 126 | # [ "name", "units", default, [lower, upper], "type","description" ], |
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| 127 | [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", "Barbell scattering length density" ], |
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| 128 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "","Solvent scattering length density" ], |
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| 129 | [ "bell_radius", "Ang", 40, [0, inf], "volume","Spherical bell radius" ], |
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| 130 | [ "radius", "Ang", 20, [0, inf], "volume","Cylindrical bar radius" ], |
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| 131 | [ "length", "Ang", 400, [0, inf], "volume","Cylinder bar length" ], |
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| 132 | [ "theta", "degrees", 60, [-inf, inf], "orientation","In plane angle" ], |
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| 133 | [ "phi", "degrees", 60, [-inf, inf], "orientation","Out of plane angle" ], |
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| 134 | ] |
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| 135 | |
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| 136 | source = [ "lib/J1.c", "lib/gauss76.c", "barbell.c" ] |
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| 137 | |
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| 138 | def ER(radius, length): |
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[1e11735] | 139 | return 1.0 |
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[58f41fe] | 140 | |
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| 141 | # parameters for demo |
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| 142 | demo = dict( |
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| 143 | scale=1, background=0, |
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| 144 | sld=6, solvent_sld=1, |
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| 145 | bell_radius = 40, radius=20, length=400, |
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| 146 | theta=60, phi=60, |
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[0706431] | 147 | radius_pd=.2, radius_pd_n=5, |
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| 148 | length_pd=.2,length_pd_n=5, |
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[58f41fe] | 149 | theta_pd=15, theta_pd_n=0, |
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| 150 | phi_pd=15, phi_pd_n=0, |
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| 151 | ) |
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| 152 | |
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| 153 | # For testing against the old sasview models, include the converted parameter |
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| 154 | # names and the target sasview model name. |
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| 155 | oldname='BarBellModel' |
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| 156 | oldpars=dict(sld='sld_barbell', |
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| 157 | solvent_sld='sld_solv', bell_radius='rad_bell', |
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| 158 | radius='rad_bar',length='len_bar') |
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